Time: Dec.16, 11:10-12:00
### The Lang-Trotter conjecture for CM elliptic curves

#### Hourong Qin

**Abstract**: Let $E$ be an elliptic curve over $\mathbb{Q}.$
For a fixed integer $r,$ define the prime-counting function
$\pi_{E,r}(x):=\sum_{p\leq x, a_E,a_p=r}1$.
If $r=0$, then assume additionally that $E$ has no
complex multiplication. The Lang-Trotter conjecture predicts that
$$\pi_{E,r}(x)=C_{E,r}\cdot \frac{\sqrt{x}}{{\rm log}x}+o\left(\frac{\sqrt{x}}{{\rm log}x}\right)$$
as $x\longrightarrow \infty,$ where $C_{E,r}$ is a specific
non-negative constant.

It is open whether there exists a polynomial in one variable of degree $2$ that represents infinitely many primes. For example, at present, we do not know whether the polynomial
$x^2+1$ represents infinitely many primes. The Hardy-Littlewood conjecture gives a similar asymptotic formula as above for the
number of primes of the form $ax^2+bx+c$.

We establish a relationship between the Hardy-Littlewood conjecture and the Lang-Trotter conjecture for CM elliptic curves.